HAL Id: hal-02373898
https://hal.archives-ouvertes.fr/hal-02373898
Submitted on 21 Nov 2019
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
EXTREMAL STRUCTURE AND DUALITY OF LIPSCHITZ FREE SPACES
Luis García-Lirola, Colin Petitjean, Antonin Prochazka, Abraham Rueda Zoca
To cite this version:
Luis García-Lirola, Colin Petitjean, Antonin Prochazka, Abraham Rueda Zoca. EXTREMAL STRUCTURE AND DUALITY OF LIPSCHITZ FREE SPACES. Mediterranean Journal of Mathe- matics, Springer Verlag, 2018, 15 (2), pp.69. �10.1007/s00009-018-1113-0�. �hal-02373898�
FREE SPACES
LUIS GARC´IA-LIROLA, COLIN PETITJEAN, ANTON´IN PROCH ´AZKA, AND ABRAHAM RUEDA ZOCA
Abstract. We analyse the relationship between different extremal notions in Lipschitz free spaces (strongly exposed, exposed, preserved extreme and extreme points). We prove in particular that every preserved extreme point of the unit ball is also a denting point. We also show in some particular cases that every extreme point is a molecule, and that a molecule is extreme whenever the two points, sayxandy, which define it satisfy that the metric segment [x, y] only containsxandy. The most notable among them is the case when the free space admits an isometric predual with some additional properties. As an application, we get some new consequences about norm attainment in spaces of vector-valued Lipschitz functions.
1. Introduction
The Lipschitz free space F(M) of a metric space M (also known as Arens- Eells space) is a Banach space such that every Lipschitz function on M admits a canonical linear extension defined on F(M) (see below for details). This fundamental linearisation property makes of Lipschitz free spaces a precious magnifying glass to study Lipschitz maps between metric spaces, and for example it relates some well-known open problems in the Banach space theory to some open problems about Lipschitz free spaces (see [15]). A considerable effort to study the linear structure and geometry of these spaces has been undergone by many researchers in the last two or three decades.
Date: September, 2017.
2010 Mathematics Subject Classification. Primary 46B20; Secondary 54E50.
Key words and phrases. Extreme point; Dentability; Lipschitz free; Duality; Uniformly discrete.
The research of L. Garc´ıa-Lirola was supported by the grants MINECO/FEDER MTM2014- 57838-C2-1-P and Fundaci´on S´eneca CARM 19368/PI/14.
The research of C. Petitjean and A. Proch´azka was supported by the French “Investissements d’Avenir” program, project ISITE-BFC”.
The research of A. Rueda Zoca was supported by a research grant Contratos predoctorales FPU del Plan Propio del Vicerrectorado de Investigaci´on y Transferencia de la Universidad de Granada, by MINECO (Spain) Grant MTM2015-65020-P and by Junta de Andaluc´ıa Grants FQM-0185.
1
In the present paper we focus on the extremal structure of F(M). The study of extremal structure plays an important role in optimisation (indeed we obtain some consequences in norm attainment of Lipschitz maps). It has probably started in [26], where it is proved that preserved extreme points of the unit ball BF(M) are always molecules (i.e. measures of the form mxy :=
(δ(x)−δ(y))/d(x, y)). Recently Aliaga and Guirao pushed further this work (see [2]). In particular, answering a question of Weaver, they showed in the compact case that the extreme points are in fact preserved, and are exactly the molecules mxy for which there are no points except xand y in the metric segment [x, y]. They also give a metric characterisation of preserved extreme points in full generality, which we prove also here by a different argument. More results in the same line appeared in [12], where a metric characterisation of the strongly exposed points is given.
However, the two main questions in this domain remain open:
a) If µ∈ext(BF(M)), is µnecessarily of the formµ= mxy for somex6=y ∈M?
b) If the metric segment [x, y] does not contain any other point of M than x and y, ismxy an extreme point of BF(M) ?
The goal of the present article is to continue the effort in exploring the extremal structure of F(M) and provide affirmative answers to both previous questions a) and b) in some particular cases. For instance, we prove that for the following chain of implications
strongly exposed=(1)⇒denting=(2)⇒preserved extreme=(3)⇒extreme, the converse of (2) holds true in general (Theorem 2.4) but the converse of (1) and (3) are both false (Examples 6.4 and 5.4 respectively). However, some of the previous implications are equivalences in some special classes of metric spaces. The most notable among them is the case when F(M) admits an isometric predual with some additional properties. We are thus led to the study of preduals of free spaces which seems interesting on its own (see Section 3).
The paper is organised as follows. In Section 2 we prove that every preserved extreme point of BF(M) is also a denting point in full generality (Theorem 2.4) and we provide a different proof of the metric characterisation of preserved extreme points given in [2]. We also show that the canonical image δ(M) of M inside F(M) as well as the set of molecules joined with 0 are weakly closed in F(M) (Proposition 2.9 and Proposition 2.13 respectively). Next in Section 3, based on ([7, 20, 26]), we study under what circumstances F(M) is isometric to a dual space. We also pin down a distinguished class of preduals, called natural preduals (see Definition 3.1), which turns out to be of particular interest in the later sections. In Section 4, we study the extremal structure of spaces admitting such a natural predual. In particular, we show under an additional
assumption that the set of extreme points coincides with the set of strongly exposed points (Corollary 4.2). Then in Section 5 we focus on the case when M is uniformly discrete and bounded. Under this assumption, the question b) has an affirmative answer (Proposition 5.1), the implication (1) admits a converse (Proposition 5.3) , and the question a) has also an affirmative answer if moreover F(M) admits a natural predual (Proposition 5.5). In Section 6 we show that the converse of (3) holds for certain compact spaces since the norm of F(M) turns out to be weak* asymptotically uniformly convex. To finish, in Section 7 we apply our work to deduce results about norm attainment of Lipschitz functions.
Notation. Throughout the paper we will only consider real Banach spaces.
Given a Banach space X, we will denote by BX (respectively SX) the closed unit ball (respectively the unit sphere) of X. We will also denote by X∗ the topological dual of X. By a slice of the unit ball BX we will mean a set of the following form
S(f, α) :={x∈BX :f(x)>1−α}, f ∈SX∗, α >0.
The notations ext(BX), exp(BX), and strexp(BX) stand for the set of extreme, exposed, and strongly exposed points of BX, respectively (we refer to [5] for formal definitions and background on this concepts). A point x∈BX is said to be a denting point of BX if BX admits arbitrarily small slices containingx.
Given a norming subspace Y ofX, we denote by σ(X, Y) the topology onX of pointwise convergence on elements of Y. Given a topological space (T, τ), we denote Cτ(T) the space of continuous functions on T.
Given a metric space M, B(x, r) denotes the closed ball in M centered at x ∈ M with radius r. Given x, y ∈ M, we denote [x, y] the metric segment between x and y, that is,
[x, y] ={z ∈M :d(x, z) +d(z, y) =d(x, y)}.
We will denote by Lip0(M, X) (or simply Lip0(M) if X = R) the space of all X-valued Lipschitz functions on M which vanish at a designated origin 0∈M. We will consider the norm in Lip0(M, X) given by the best Lipschitz constant, denotedk·kL. Of particular interest to us is the space oflittle-Lipschitz functions,
lip0(M) :=
(
f ∈Lip0(M) : lim
ε→0 sup
0<d(x,y)<ε
|f(x)−f(y)|
d(x, y) = 0 )
.
We denoteδ the canonical isometric embedding of M intoF(M), which is given by hf, δ(x)i =f(x) for x∈ M and f ∈Lip0(M). By a molecule we mean an
element of F(M) of the form
mxy := δ(x)−δ(y) d(x, y)
for x, y ∈M,x6= y. The set of all molecules in M will be denoted by V. Note by passing that V is a norming set for Lip0(M) and soBF(M) = co(V). We will need for every x, y ∈M, x6=y, the function
fxy(t) := d(x, y) 2
d(t, y)−d(t, x) d(t, y) +d(t, x).
The properties collected in the next lemma have already been proved in [18].
They make of fxy a useful tool for studying the geometry of BF(M). Lemma 1.1. Let x, y ∈M with x6=y. We have
(a) fxy(u)−fd(u,v)xy(v) ≤ max{d(x,u)+d(u,y),d(x,v)+d(v,y)}d(x,y) for all u6=v ∈M. (b) fxy is Lipschitz and kfxykL≤1.
(c) Let u6=v ∈M and ε >0 be such that fxy(u)−fd(u,v)xy(v) >1−ε. Then (1−ε) max{d(x, v) +d(y, v), d(x, u) +d(y, u)}< d(x, y).
(d) If u6=v ∈M and fxy(u)−fd(u,v)xy(v) = 1, then u, v ∈[x, y].
2. General results
Our first goal is to show that every preserved extreme point of BF(M) is also a denting point. In order to prove that result we need the following characterisation of preserved extreme points which appears in [16] and that we state for future reference.
Proposition 2.1 (Proposition 9.1 in [16]). Let X be a Banach space, C be a closed bounded convex subset of X and x ∈ C. Then, the following are equivalent:
(i) x is an extreme point of Cω
∗
the weak*-closure of C in X∗∗.
(ii) The slices of C containingx are a neighbourhood basis of x for the weak topology in C.
(iii) For every sequences {yn} and {zn} in C such that yn+z2 n →k·k x we have that yn →w x.
It is easy to check that conditions above are also equivalent to the following:
(iii’) For every λ ∈ (0,1) and sequences {yn} and {zn} in C such that λyn+ (1−λ)zn→k·k x we have thatyn, zn →w x.
Next lemma asserts that a net of molecules which converges to a molecule in the weak topology in fact converges in the norm topology. This lemma will be useful in the proof of Theorem 2.4 and also in order to show that the set of molecules is not far from being weakly closed (Proposition 2.13).
Lemma 2.2. Assume {mxαyα} is a net in V which converges weakly to mxy. Then limαd(xα, x) = 0 and limαd(yα, y) = 0.
Proof. Assume that 0< ε <min{d(x, y),lim supαd(xα, x)}. Consider the map f given by f(t) = (ε−d(x, t))+ and let g = f −f(0)∈ Lip0(M). Note that hg, mxyi= d(x,y)ε >0. However,
lim inf
α hg, mxαyαi= lim inf
α
−f(yα) d(xα, yα) ≤0,
a contradiction. Therefore, limαxα = x. Analogously we get that limαyα =
y.
We need the following variation of Asplund–Bourgain–Namioka superlemma [5, Theorem 3.4.1].
Lemma 2.3. Let A, B be bounded closed convex subsets of a Banach space X and let ε >0. Assume that diam(A) < ε and that there is x0 ∈ A\B which is a preserved extreme point of co(A∪B). Then there is a slice of co(A∪B) containing x0 which is of diameter less than ε.
Proof. For each r∈[0,1] let
Cr={x∈X :x= (1−λ)y+λz, y∈A, z ∈B, λ∈[r,1]}.
The proof of the Superlemma says that there isrso that diam(co(A∪B)\Cr)< ε.
We will show thatx0 ∈/ Cr. Thus, any slice separatingx0 from Cr will do the work. To this end, assume that there exist sequences {yn} ⊂A, {zn} ⊂B and λn⊂[r,1] such that x0 = limn(1−λn)yn+λnzn. By extracting a subsequence, we may assume that {λn} converges to some λ∈ [r,1]. Note that then x0 = limn(1−λ)yn+λzn. Since x0 is a preserved extreme point, this implies that {zn}converges weakly tox0 by Proposition 2.1. That is impossible sincex0 ∈/ B and B is weakly closed as being convex and closed.
Theorem 2.4. Let M be a pointed metric space. Then every preserved extreme point of BF(M) is a denting point.
Proof. Let µbe a preserved extreme point of BF(M), which must be an element of V. Denote by S the set of weak-open slices of BF(M) containing µ. Consider the order S1 ≤S2 ifS2 ⊂S1 forS1, S2 ∈ S. Using (ii) of Proposition 2.1, every finite intersection of elements of S contains an element ofS and so (S,≤) is a
directed set. Assume that µis not a denting point. Then, there is ε >0 so that diam(S)>2ε for every S∈ S.
We distinguish two cases. Assume first that for every slice S of BF(M) there isµS ∈(V ∩S)\B(µ, ε/4). Then{µS}is a net inV which converges weakly to µ. By Lemma 2.2, it also converges in norm, which is impossible. Thus, there is a slice S of BF(M) such that diam(V ∩S)< ε/2. Note that
BF(M)= co(V) = co(co(V ∩S)∪co(V \S))
and so the hypotheses of Lemma 2.3 are satisfied for A = co(V ∩S), B = co(V \S), and µ∈A\B (taking the closed convex hull does not change the diameter). Then there is a slice of BF(M) containing µ of diameter less than ε,
a contradiction.
Theorem 2.4 provides a new proof of the following result given in [12].
Corollary 2.5. Let M be a length space. Then BF(M) does not have any preserved extreme point.
Proof. The space F(M) has the Daugavet property whenever M is a length space [17]. In particular, every slice of BF(M) has diameter two. Thus,BF(M)
does not have any denting point.
During the preparation of this paper we have learnt that Aliaga and Guirao [2]
characterised metrically the preserved extreme points of free spaces. In the following pages we provide an alternative proof of their result which accidentally reproves our Theorem 2.4.
Theorem 2.6. Let M be a pointed metric space andx, y ∈M. The following are equivalent:
(i) The molecule mxy is a denting point of BF(M).
(ii) For every ε >0 there exists δ >0 such that every z ∈M satisfies (1−δ)(d(x, z) +d(z, y))< d(x, y) =⇒min{d(x, z), d(y, z)}< ε.
Proof of (i)⇒(ii). In fact we are going to show that negation of (ii) implies that mxy is not a preserved extreme point. Since denting points are trivially preserved extreme points, this will show at once that mxy is not denting.
So let us fix ε >0 such that for every n∈N there exists zn∈M such that
1− 1 n
(d(x, zn) +d(zn, y))< d(x, y)
but min{d(x, zn), d(y, zn)} ≥ ε. Let µ be a w∗-cluster point of {zn} ({zn} is clearly bounded). By lower semicontinuity of the norm we have
kδ(x)−µk+kµ−δ(y)k=d(x, y).
If µ ∈ {δ(x), δ(y)}, say µ = δ(x) then by Lemma 2.2 we get that zn → x in (M, d) which is a contradiction.
Thus µ /∈ {δ(x), δ(y)}. Then δ(x)−δ(y)
kδ(x)−δ(y)k = kδ(x)−µk kδ(x)−δ(y)k
δ(x)−µ
kδ(x)−µk+ kµ−δ(y)k kδ(x)−δ(y)k
µ−δ(y) kµ−δ(y)k. Thusµ is a non-trivial convex combination and so it is not preserved extreme
which concludes the proof of (i)⇒(ii).
For the proof of the other implication we need a couple of lemmata. The first of them shows that the diameter of the slices of the unit ball can be controlled by the diameter of the slices of a subset of the ball that is norming for the dual.
Lemma 2.7. LetXbe a Banach space and letV ⊂SX be such thatBX = co(V).
Let f ∈BX∗ and 0< α, ε <1. Then
diam(S(f, εα))≤2 diam(S(f, α)∩V) + 4ε.
Proof. Fix a point x0 ∈ S(f, α) ∩V. It suffices to show that kx −x0k <
diam(S(f, α)∩V) + 2ε for everyx∈S(f, εα)∩co(V). To this end, letx∈BX be such that f(x)>1−εα, and x=Pn
i=1λixi, with xi ∈V, Pn
i=1λi = 1 and λi >0 for all 1≤i≤n. Define
G={i∈ {1, . . . , n}:f(xi)>1−α}
and B ={1, . . . , n} \G. We have 1−εα < f(x) =X
i∈G
λif(xi) +X
i∈B
λif(xi)
≤X
i∈G
λi+ (1−α)X
i∈B
λi = 1−αX
i∈B
λi,
which yields thatP
i∈Bλi < ε. Now, kx−x0k ≤X
i∈G
λikxi−x0k+X
i∈B
λikxi−x0k ≤diam(S(f, α)∩V) + 2ε.
Lemma 2.8. Letx, y ∈M,x6= y, such thatd(x, y) = 1. For every0< ε <1/4 and 0< τ <1 there is a function f ∈Lip0(M) such that kfkL = 1, hf, mxyi>
1−4ετ and satisfying that for every u, v ∈ M, u 6= v, if u, v ∈ B(x, ε) or u, v ∈B(y, ε), then hf, muvi ≤1−τ.
Proof. Definef: B(x, ε)∪B(y, ε)→R by f(t) =
( 1
1+4ετ(τ + (1−τ)d(y, t)) if t∈B(x, ε),
1
1+4ετ(1−τ)d(y, t) if t∈B(y, ε).
Note that
hf, mxyi=f(x)−f(y) = 1
1 + 4ετ >1−4ετ.
Moreover, note that if u, v ∈B(x, ε) or u, v ∈B(y, ε) then hf, muvi ≤ 1+4ετ1−τ ≤ 1−τ, so the last condition in the statement is satisfied. Now we compute the Lipschitz norm of f. It remains to compute hf, muvi with u ∈ B(x, ε) and v ∈B(y, ε). In that case we have
|hf, muvi|= |τ + (1−τ)(d(u, y)−d(v, y))|
(1 + 4ετ)d(u, v) ≤ τ + (1−τ)d(u, v) (1 + 4ετ)d(u, v)
≤ 1
1 + 4ετ τ
1−2ε + 1−τ
≤ τ(1 + 4ε) + 1−τ 1 + 4ετ = 1
where we are using that (1−2ε)−1 ≤1 + 4ε since ε <1/4. This shows that kfkL≤1. Next, find an extension of f with the same norm. Finally, replace f
with the function t7→f(t)−f(0).
Proof of (ii)⇒(i) of Theorem 2.6. Now, assume that (ii) holds. We can assume that d(x, y) = 1. Fix 0 < ε < 1/4. We will find a slice of BF(M) containing mxy of diameter smaller than 32ε. Let δ >0 be given by property (ii), clearly we may assume that δ <1. Let f be the function given by Lemma 2.8 with τ =δ/2. Define
h(t) := fxy(t) +f(t)
2 .
It is clear that khkL ≤1. Moreover, note that hh, mxyi= hfxy, mxyi+hf, mxyi
2 >1−2ετ = 1−εδ.
Take α= δ/4 and consider the slice S =S(h, α). Note that mxy ∈S(h,4εα).
We will show that diam(S∩V)≤8ε and as a consequence of Lemma 2.7 we will get that diamS(h, α)≤32ε. So letu, v ∈M be such that muv ∈S. First, note that hfxy, muvi>1−δ, since otherwise we would have
hh, muvi= 1
2(hfxy, muvi+hf, muvi)≤ 1
2(1−δ) + 1
2 = 1− δ
2 <1−α.
Thus, from the property (c) of the functionfxy in Lemma 1.1 and the hypothesis (ii) we have that
min{d(x, u), d(u, y)}< ε and min{d(x, v), d(y, v)}< ε.
On the other hand,
1−α <hh, muvi ≤ 1 2+ 1
2hf, muvi
and so hf, muvi > 1 −2α = 1 − δ2 = 1− τ. Thus, we have that u and v do not belong simultaneously to neither B(x, ε) nor B(y, ε). If d(x, v) < ε and d(y, u) < ε, then it is easy to check that hfxy, muvi ≤ 0. So necessarily d(x, u)< ε and d(y, v)< ε. Now, use the estimate
kmxy−muvk= kd(u, v)(δ(x)−δ(y))−d(x, y)(δ(u)−δ(v))k d(x, y)d(u, v)
≤ k(δ(x)−δ(y))−(δ(u)−δ(v))k
d(x, y) +|d(u, v)−d(x, y)|kδ(u)−δ(v)k d(x, y)d(u, v)
≤2d(x, u) +d(y, v) d(x, y) ≤4ε.
Therefore, diam(S∩V)≤8ε.
2.1. Weak topology in free spaces. The results which follow are independent of the rest of the article. The reader interested only in the extremal structure of the free spaces can skip until Section 3.
Simple examples (Examples 5.6 and 5.7 ) show that δ(M) is not necessarily weak∗ closed when F(M) is a dual space. The next proposition shows that the situation is different for the weak topology.
Proposition 2.9. Let M be a complete pointed metric space. Then δ(M) ⊂ F(M) is weakly closed.
The proposition could be deduced more or less easily from Proposition 2.1.6 in [26] but we propose a self-contained proof. For the proof we will need the next observation (essentially already present in [26]). The weak*-closures of subsets of F(M) below are taken in the bidualF(M)∗∗= Lip0(M)∗.
Lemma 2.10. Let M be a complete pointed metric space. Let µ∈ δ(M)w
∗
\ δ(M). Then there exists ε >0 such that for all q1, . . . , qn∈M we have that
µ∈δ M \
n
[
i=
B(qi, ε)
!w
∗
.
Proof. Indeed, otherwise we could find a sequence {qn} ⊂ M such that µ ∈ δ(B(qn,2−n))w
∗
for every n≥1. It follows that kµ−δ(qn)k ≤2−n for every n and thus {qn}is Cauchy. By completeness ofM it follows thatµ= limnδ(qn)∈
δ(M). This contradiction proves the claim.
Proof of Proposition 2.9. It is enough to show that if µ ∈ δ(M)w
∗
\ δ(M), then µ is not w∗-continuous. Indeed, this yields that µ /∈ F(M) and so δ(M)w =δ(M)w∗∩ F(M) =δ(M).
So let µ ∈ δ(M)w∗ \ δ(M) and let ε > 0 be as in Lemma 2.10. Now let U be an open neighbourhood of 0 in (BLip
0(M), w∗). Since the w∗ topol- ogy and the topology of pointwise convergence coincide on the ball BLip0(M), we may assume that there are x1, . . . , xn ∈ M and α > 0 such that U = f ∈BLip0(M) :|f(xi)|< αfor i= 1, . . . n . We definef(x) := dist(x,{x1, . . . , xn}).
We clearly havef ∈U. Moreover since µ∈δ M \
n
[
i=1
B(xi, ε)
!w
∗
,
we have thatµ(f)≥ε. Thus µis not weak*-continuous asU was arbitrary.
We observe the following curious corollary (which also admits an independent proof by combinatorial methods).
Corollary 2.11. Let M be a complete pointed metric space. If {xn} ⊂M is a sequence such that δ(xn) converges weakly to someµ∈ F(M), then there exists x∈M such that µ=δ(x) and d(xn, x)→0.
Proof. The fact that µ=δ(x) follows from Proposition 2.9. For the rest it is enough to pose f(·) := d(·, x)−d(0, x) and use that d(xn, x) = hδ(xn), fi −
hδ(x), fi →0.
Given a complete metric space M and µ ∈ F(M)\δ(M) there is a weak neighbourhood that separates µ from δ(M). The next example shows that contrary to what one might expect, such a neighbourhood is not necessarily of the form {γ ∈ F(M) :|hf, γ−µi|< ε} for some f ∈Lip0(M) and ε >0.
Example 2.12. LetM = [0,1] with the usual metric and let µbe the Lebesgue measure on [0,1]. It is well known and can be easily shown using the Riemann sums that µ ∈ F(M). It acts on Lip0([0,1]) as follows hµ, fi = R1
0 f(t)dt.
Now the mean value theorem implies that for every f ∈Lip0(M) there exists x∈[0,1] such that hδ(x), fi=hµ, fi.
In light of Proposition 2.9, it is also natural to wonder if the setV of molecules is weakly closed. It is known that 0 is in the weak-closure of V whenever M is not bi-Lipschitz embeddable in RN (see Lemma 4.2 in [13]). The following proposition shows that 0 is the only point that we can reach taking the weak- closure of V. On the other hand, 0 is never in the sequential closure ofV which we will show in a corollary below.
Proposition 2.13. Let(M, d) be a complete pointed metric space. Then Vw ⊂ V ∪ {0}.
Proof. The proof is based on [26, Theorem 2.5.3]. Let us begin with an expla- nation of this result. To this end, let ˜M :={(x, y)∈M2 : x6=y} and
Φ : Lip0(M)→ Cb( ˜M) f(x, y)7→ f(x)−f(y)
d(x, y)
(here Cb( ˜M) stands for the continuous and bounded functions on ˜M). It is easy to see that Φ is an isometry. Now let us denote βM˜ the Stone- ˇCech compactification of ˜M. As usual, we can canonically identify Cb( ˜M) with C(βM˜) so that we now see Φ as a map from Lip0(M) to C(βM˜). Thus Φ∗ goes from C(βM˜)∗ = M(βM˜) to Lip0(M)∗. According to Weaver, we say that µ∈Lip0(M)∗ is normal if{hµ, fii} converges to hµ, fiwhenever {fi} is a bounded and decreasing (meaning that fi ≥fj fori≤j) net in Lip0(M) which w∗-converges to f ∈Lip0(M). Clearly normality is implied by w∗-continuity.
Finally, [26, Theorem 2.5.3] asserts that if x ∈ βM˜ with Φ∗δ(x) 6= 0, then Φ∗δ(x) is normal if and only if x∈M˜.
Let us now prove the assertion of the proposition. Since Vw =Vw
∗
∩ F(M) ={µ∈Vw
∗
: µ isw∗-continuous}, it is enough to show that if µ∈ Vw
∗
\(V ∪ {0}) then µ is not w∗-continuous.
So let us fix such a µ. We identify, as we may, ˜M with δ( ˜M)⊂ M(βM˜). We claim that δ( ˜M) is homeomorphic to (V, w∗). Indeed, it is clear that
Φ∗δ( ˜M) :δ(x, y)∈δ( ˜M)7→mxy ∈(V, w∗)
is continuous and bijective. The fact that the inverse mapping is also continuous follows from Lemma 2.2. So the claim is proved. Now (Vw
∗
, w∗) is clearly a compactification of V. Thus the universal property of the Stone- ˇCech compacti- fication provides a surjective extension of Φ∗δ( ˜M) that goes fromδ(βM˜) toVw
∗
. It is easy to check that the latter extension is in fact Φ∗δ(βM)˜ :δ(βM˜)→Vw
∗
. Now consider x∈βM˜ such that Φ∗δ(x) =µ6= 0. Since µ∈Vw
∗
\(V ∪ {0}), we deduce that x6∈M˜. Thus, according to [26, Theorem 2.5.3], Φ∗δ(x) =µ is not normal and therefore not w∗-continuous. This ends the proof.
From the previous proposition, we deduce a result similar to Corollary 2.11.
Corollary 2.14. LetM be a complete pointed metric space. If{µn} ⊂ F(M)is a sequence of molecules (µn=mxnyn) which converges weakly to someµ∈ F(M), then there exist x6=y∈M such that µ=mxy and {µn} actually converges in norm to mxy. In particular, a sequence of molecules cannot converge weakly to 0 and so V is weakly sequentially closed.
Proof. Proposition 2.13 shows that µ = mxy or µ = 0. In the first case the sequence {µn} actually converges in norm by Lemma 2.2.
If µ= 0 then clearly {µn} does not admit any norm convergent subsequence.
Therefore it is not totally bounded and so there existε >0 and a subsequence (nk)⊂N such that kµnk−µnlk ≥ε for all k 6=l.
Now {µnk} is a uniformly separated bounded sequence of measures such that the cardinality of their supports is bounded. So the deep Theorem 5.2 in [1]
shows that {µnk} cannot converge weakly to 0 which is a contradiction.
Now that we know that Vw ⊂ V ∪ {0} we get an easy proof of Weaver’s theorem [26] which claims that the preserved extreme points are molecules. We include it for completeness as it is directly related to the main subject of this paper.
Corollary 2.15. Let M be a complete pointed metric space and let µ be a preserved extreme point of BF(M). Then µ=mxy for some x6=y∈M.
Proof. Indeed, we have that co(V) = BF(M) and BF(M) w∗
= BLip0(M)∗. Thus cow∗(V) =BLip0(M)∗ and so by Milman’s theorem ext(BLip0(M)∗)⊂Vw
∗
. Finally we get thatF(M)∩ext(BLip0(M)∗)⊂Vw and so Proposition 2.13 yieldsF(M)∩
ext(BLip0(M)∗)⊂V.
3. Duality of some Lipschitz free spaces
Many of our results in Sections 4 and 5 use the hypothesis that F(M) admits an isometric predual which makes δ(M) w∗-closed. Even though for some of these results we do not know whether this hypothesis is superfluous, we take the opportunity to study the Lipschitz free spaces which admit such a predual.
Definition 3.1. Let M be a bounded pointed metric space. We will say that a Banach space X is a natural predual of F(M) ifX∗ =F(M) isometrically and δ(M) is σ(F(M), X)-closed.
It is obvious that when M is a compact metric space then every isometric predual of F(M) is natural. We will show in Examples 5.6 and 5.7 that there are isometric preduals to F(M) which are not natural.
Let us state for the future reference an almost obvious characterisation of natural preduals.
Proposition 3.2. Let M be a bounded pointed metric space and let X be an isometric predual of F(M). Then the following are equivalent:
(i) There is a compact Hausdorff topologyτ on M such thatX ⊂Lip0(M)∩ Cτ(M).
(ii) δ(M) is σ(F(M), X)-closed.
Proof. We only need to show (i)⇒(ii). To this end, note that the w∗-topology of F(M) and the τ-topology coincide on δ(M). Indeed, everyw∗-open set in δ(M) is also τ-open since X is made up of τ-continuous functions, so that the w∗-topology is weaker than τ on δ(M). By compactness of the Hausdorff
topology τ, we have that they agree on δ(M).
The natural preduals are quite common. In fact, the known constructions of isometric preduals to F(M) when M is bounded all produce natural preduals.
Indeed, this is the case for Theorem 3.3.3 in [26] as well as Theorem 2.1 in [7]
because of the compactness. In the next theorem we will show that it is also true for Theorem 6.2 in [20]. We will say that a subspace X of Lip0(M) 1-separates points uniformly (shortened 1-S.P.U.) if for every x, y ∈ M and every ε > 0 there is f ∈X such that f(x)−f(y) =d(x, y) and kfkL <1 +ε.
Proposition 3.3 (Theorem 6.2 in [20]). Let M be a separable bounded pointed metric space and let τ be a topology on M so that (M, τ) is compact. Assume that X = lip0(M)∩ Cτ(M) 1-S.P.U. Then X is a natural predual of F(M).
In what follows we provide a slightly different proof of Kalton’s result, based now on Petun¯ın-Pl¯ıˇcko theorem (see [14, 23]). We recall that this last theorem asserts that a closed subspace S ⊂X∗ of the dual of a separable Banach space X is an isometric predual of X (that is S∗ =X) if, and only if,S is composed of norm-attaining functionals and S separates the points of X. The use of this theorem to produce preduals to free spaces has become quite common (see [6, 7, 8, 11] and also our Examples 5.6 and 5.7). The benefit of this proof is that it avoids the metrizability assumption of the topologyτ present in Kalton’s original exposition of this result.
In the proof we will also need the following lemma which restates in a general framework the first step of Kalton’s proof.
Lemma 3.4. Let(M, d)be a pointed metric space such that there is a topology τ on M and a subsetX ⊂Lip0(M)∩ Cτ(M)which 1-S.P.U. Then d: (M, τ)2 →R is l.s.c.
Proof. Let {xα}, {yα} be τ-convergent nets in M with limits x andy, respec- tively. Given ε > 0, find f ∈ X such that f(y)−f(x) ≥ d(x, y) −ε and kfkL= 1. Then
d(x, y)−ε≤lim
α f(yα)−f(xα)≤lim inf
α d(xα, yα)
and the arbitrariness of ε yields the desired conclusion.
Proof of Proposition 3.3. First of all, according to Lemma 3.4, note that d is τ-l.s.c.
Now, we need to verify the conditions of Petun¯ın and Pl¯ıˇcko’s theorem. First, since M is bounded, we see that S is a closed subspace of Lip0(M). Second, S is separating since it is a lattice and separates the points of M uniformly (see [20, Proposition 3.4]).
Finally it remains to show that X is made of norm-attaining functionals.
To this end, let f ∈ SX and take sequences {xn}, {yn} in M such that limnf(xd(xn)−f(yn)
n,yn) = 1. Note that infnd(xn, yn) =: θ > 0 since f ∈ lip0(M).
By the compactness of (M, τ) and the boundedness of d, we can find subnets {xα}of{xn}and{yα}of{yn}such thatxα →τ x,yα→τ yandd(xα, yα)→C > 0.
Then,
1 = lim
α
f(xα)−f(xα)
d(xα, yα) → f(x)−f(y)
C ≤ f(x)−f(y) d(x, y) . ThusX is made up of norm-attaining functionals.
To conclude, we get that S is a natural predual by just applying Proposi-
tion 3.2.
The next proposition testifies that Kalton’s theorem is the only way to build a natural predual if the predual is moreover required to be a subspace of little Lipschitz functions.
Proposition 3.5. LetM be a bounded pointed metric space and letX∗ = F(M) be a natural predual such that X ⊆ lip0(M). Then there exists a topology τ on M such that (M, τ) is compact, the metric d: (M, τ)2 → R is l.s.c. and X = lip0(M)∩ Cτ(M).
Proof. We put τ := {δ−1(U) :U ∈σ(F(M), X)}. Since δ(M) is σ(F(M), X)- closed and bounded, (M, τ) is compact. Recall that d(x, y) = kδ(x)−δ(y)k and k·kis σ(F(M), X)-lsc, so the metricd is τ-lsc. Since
X ={x∗ ∈ F(M)∗ :x∗ is σ(F(M), X)−continuous}
and X ⊂ lip0(M), we get that X ⊆ lip0(M, d)∩ Cτ(M) =: Y. This means that σ(F(M), Y) is stronger than σ(F(M), X). On the other hand, Propo- sition 3.3 yields that Y∗ = F(M). Therefore, by compactness, σ(F(M), X) and σ(F(M), Y) coincide on BF(M). As a consequence of Banach-Dieudonn´e theorem, they coincide on F(M). This means that
X ={x∗ ∈ F(M)∗ :x∗ is σ(F(M), X)−continuous}
={x∗ ∈ F(M)∗ :x∗ is σ(F(M), Y)−continuous}=Y.
But one should be aware that not all natural preduals are contained in the space of little Lipschitz functions.
Example 3.6. Let M =1
n :n∈N ∪ {0} with the distance comming from the reals. Then it is well known that F(M) is isometrically isomorphic to `1. Further we know (Theorem 2.1 in [7]) that lip0(M) is isometrically a predual.
Since M is compact, every predual is natural. So our Proposition 3.5 and the fact that M is compact show that any isometric predual of `1 which is not isometric to lip0(M) intersects the complement of lip0(M).
Note that Lip0(M) = lip0(M) whenM is uniformly discrete. This observation and the previous results yield the following corollary.
Corollary 3.7. Let (M, d) be a uniformly discrete bounded separable pointed metric space with 0∈M. Let X be a Banach space. Then it is equivalent:
(i) X is a natural predual of F(M).
(ii) There is a Hausdorff topology τ on M such that (M, τ) is compact, d is τ-l.s.c. and X = Lip0(M, d)∩ Cτ(M) equipped with the norm k·kL. Proof. (ii)⇒(i) Given x, y ∈ M, x 6= y, define f: {x, y} → R by f(x) = 0 and f(y) = d(x, y). By Matouskova’s extension theorem [21], there is ˜f ∈ Lip0(M)∩ Cτ(M) extending f such that
f˜
L
= 1. Thus, the hypotheses of Proposition 3.3 are satisfied.
The implication (i)⇒(ii) is contained in Proposition 3.5.
In what follows we are going to develop yet another sufficient condition for an isometric predual to be natural with the goal to show that certain preduals constructed by Weaver in [25] are natural.
Proposition 3.8. Let M be a uniformly discrete, bounded, separable, pointed metric space and let X ⊂ Lip0(M) be a Banach space such that X∗ = F(M) isometrically. If for every x∈M\ {0} the indicator function 1{x} belongs to X, then X is a natural predual of F(M). Moreover 0 is the unique accumulation point of (δ(M), w∗) and X is isomorphic toc0.
The proof will be based on the following general fact.
Lemma 3.9. Let X, Y be Banach spaces such that X∗ = Y isometrically, Y admits a bounded Schauder basis {un} and the biorthogonal functionals {u∗n} belong to X. Then un →0 weakly*.
Proof. We will show that every subsequence of {un} admits a further subse- quence that converges weakly* to 0. So let us consider such subsequence. By the weak* compactness and separability, it admits a weak* convergent subsequence, let us call it {un} again. So we haveun →u∈X weakly*. But this means that for every m∈N we have
0 = lim
n→∞hu∗m, uni=hu∗m, ui.
Thusu= 0.
Proof of Proposition 3.8. Since M is bounded and uniformly discrete, the se- quence {δ(x)}x∈M\{0} is a Schauder basis which is equivalent to the unit vector basis of `1. The biorthogonal functionals are exactly the indicator functions1{x}
for x6= 0. Applying Lemma 3.9 we get that δ(M) is weak* closed and that 0 is the unique w∗-accumulation point ofδ(M). Let τ be the restriction of thew∗- topology toM. Now Corollary 3.7 yields thatX = Lip0(M)∩Cτ(M). But, since M is bounded and uniformly discrete, we have that Lip0(M) is just all bounded functions that vanish at 0. It follows immediately that X =c0(M \ {0}).
Remark 3.10. In [25], Weaver proved a duality result forrigidly locally compact metric spaces. We recall that a locally compact metric space is said to be rigidly locally compact (see the paragraph before Proposition 3.3 in [25]) if for every r > 1 and every x ∈ M, the closed ball B(x,d(0,x)r ) is compact. The duality result of Weaver in particular implies that for a separable uniformly discrete bounded metric space M which is rigidly compact, the space
X =
f ∈Lip0(M) : f(·)
d(·,0) ∈C0(M)
is an isometric predual of F(M). Here C0(M) denotes the set of continuous functions which are arbitrarily small out of compact sets. Since it is obvious that the indicator functions 1{x} belong to X, Proposition 3.8 implies that X is a natural predual of F(M) and that X is isomorphic to c0. This shows that in the case of uniformly discrete bounded spaces, Corollary 3.7 covers the cases in which Weaver’s result ensures the existence of a predual.
Moreover, there is a metric space which satisfies the hypotheses of Corollary 3.7 and which is not rigidly locally compact.
Example 3.11. Let us consider the metric space M = {0,1} ×N equipped with the following distance: d((0, n),(1, m)) = 2 for n, m∈ N, and if n 6= m we have d((0, n),(0, m)) = 1 and d((1, n),(1, m)) = 1. Then M satisfies the assumptions of Corollary 3.7. Indeed, declare (0,1) to be the accumulation point of the sequence {(0, n)}, (1,1) to be the accumulation point of the sequence {(1, n)}, and then declare all the other points isolated. Now independently of the choice of the distinguished point 0M, M is not rigidly locally com- pact. For instance, say that 0M = (0, n). Then for every r > 1, the ball B((1,1), d(0M,(1,1))/r) =B((1,1),2/r) contains all the elements of the form (1, m) with m ∈ N. Consequently the considered ball is not compact, which
proves that M is not rigidly locally compact.
4. Extremal structure for spaces with natural preduals We are going to focus now on the extreme points in the free spaces that admit a natural predual. Assuming moreover that the predual is a subspace of little Lipschitz functions we get an affirmative answer to one of our main problems.
Note that this is an extension of Corollary 3.3.6 in [26], where it is obtained the same result under the assumption that M is compact.
Proposition 4.1. LetM be a bounded pointed metric space. Assume that there is a subspace X of lip0(M) which is a natural predual of F(M). Then
ext(BF(M))⊂V.
Proof. By the separation theorem we have that BF(M) = cow∗(V). Thus, according to Milman theorem (see [9, Theorem 3.41]), we have ext(BF(M))⊂ Vw
∗
. So let us consider γ ∈ ext(BF(M)). Take a net {mxα,yα} in V which w∗- converges to γ. By w∗-compactness of δ(M), we may assume (up to extracting subnets) that {δ(xα)} and{δ(yα)} converge to some δ(x) and δ(y) respectively.
Next, we claim that we may also assume that {d(xα, yα)} converges toC > 0.
Indeed, since M is bounded, we may assume up to extract a further subnet that {d(xα, yα)} converges toC ≥0. By assumption, there is f ∈X such that hf, γi>kγk/2 = 1/2. Sincef ∈lip0(M), there existsδ >0 such that whenever z1, z2 ∈M satisfy d(z1, z2)≤δ then we have |f(z1)−f(z2)| ≤ 12d(z1, z2). Since
limα hf, mxα,yαi=hf, γi> 1 2,
there is α0 such thathf, mxα,yαi>1/2 for everyα > α0. Thusd(xα, yα)> δ for α > α0, which implies that C ≥δ >0. Summarizing, we have a net {mxα,yα} which w∗-converges to δ(x)−δ(y)C . So, by uniqueness of the limit, γ = δ(x)−δ(y)C . Since γ ∈ext(BF(M))⊂SF(M), we get that C=d(x, y) and soγ =mxy.
We have learned that a weaker version of the following proposition appears in the preprint [2] for compact metric spaces. Our approach, which is independent of [2], also yields a characterisation of exposed points of BF(M).
Corollary 4.2. Let M be a bounded separable pointed metric space. Assume that there is a subspace X of lip0(M)which is a natural predual of F(M). Then given µ∈BF(M) the following are equivalent:
(i) µ∈ext(BF(M)).
(ii) µ∈exp(BF(M)).
(iii) There are x, y ∈M, x6=y, such that [x, y] ={x, y} and µ=mxy. Proof. (i)⇒(iii) follows from Proposition 4.1. Moreover, (ii)⇒(i) is clear, so it only remains to show (iii)⇒(ii). To this end, let x, y ∈ M, x 6=y, be so that
[x, y] ={x, y}. Consider
A={µ∈BF(M) :hfxy, µi= 1}.
We will show that A = {mxy} and so mxy is exposed by fxy in BF(M). Let µ∈ext(A). Since A is an extremal subset of BF(M),µis also an extreme point of BF(M) and so µ ∈ V ∩A. Recall that if hfxy, mu,vi = 1 then u, v ∈ [x, y], therefore V ∩A = {mxy}. Thus ext(A) ⊂ {mxy}. Finally note that A is a closed convex subset of BF(M) and so A= co(ext(A)) ={mxy} since the space
F(M) has (RNP) as being a separable dual.
It is proved in Aliaga and Guirao’s paper [2] that if (M, d) is compact, then a molecule mxy is extreme in BF(M) if and only if it is preserved extreme if and only if [x, y] ={x, y}. Thus, if lip0(M) 1-S.P.U. (and thusF(M) = lip0(M)∗), Proposition 4.1 and Aliaga and Guirao’s result provide a complete description of the extreme points: they are the molecules mxy such that [x, y] ={x, y}. It is possible to obtain the same kind of complete descriptions in some different settings as it is proved in the following result (see also Section 5).
Proposition 4.3. Let(M, d)be a bounded pointed metric space for which there is a Hausdorff topology τ such that (M, τ) is compact and d: (M, τ)2 →R is l.s.c. Let 0 < p < 1 and let (M, dp) be the p-snowflake of M. Then given µ∈BF(M) the following are equivalent:
(i) µ∈ext(BF(M,dp)).
(ii) µ∈strexp(BF(M,dp)).
(iii) There are x, y ∈M, x6=y, such that µ=mxy.
Observe that under the hypotheses above it is not necessarily true that F(M) is a dual space, but F(M, dp) already is.
Proof. (iii) =⇒ (ii). Let us fix x 6= y ∈ M. Since 0 < p < 1, it is readily seen that [x, y] ={x, y}. Moreover it is proved in [26, Proposition 2.4.5] that there is a peaking function at (x, y). Thus mxy is a strongly exposed point ([12, Theorem 4.4]). The implication (ii) =⇒ (i) is obvious. To finish, the implication (i) =⇒(iii) follows directly from Proposition 4.1 and the fact that
[x, y] ={x, y} for every x6=y ∈M.
Next we will show that the extremal structure of a free space has impact on its isometric preduals. If a metric space M is countable and satisfies the assumptions of Proposition 4.1, then ext(BF(M)) is also countable. Therefore, any isometric predual of F(M) is isomorphic to a polyhedral space by a theorem of Fonf [10], and so it is saturated with subspaces isomorphic to c0. This applies for instance in the following cases.
Corollary 4.4. Let M be a countable compact pointed metric space. Then any isometric predual of F(M)(in particular lip0(M)) is isomorphic to a polyhedral space.
Corollary 4.5. Let (M, d) be a uniformly discrete bounded separable pointed metric space such that F(M) admits a natural predual. Then any isometric predual of F(M) is isomorphic to a polyhedral space.
5. The uniformly discrete case
We have already witnessed that in the class of uniformly discrete and bounded metric spaces, many results about F(M) become simpler. Yet another example of this principle is the following main result of this section.
Proposition 5.1. Let (M, d) be a bounded uniformly discrete pointed metric space. Then a molecule mxy is an extreme point of BF(M) if and only if [x, y] ={x, y}.
Also we will need the following observation, perhaps of independent interest:
Since a point x ∈ BX is extreme if and only if x ∈ ext(BY) for every 2- dimensional subspace Y of X, the extreme points of BF(M) are separably determined. Let us be more precise.
Lemma 5.2. Assume thatµ0 ∈BF(M) is not an extreme point of BF(M). Then there is a separable subset N ⊂M such that µ0 ∈ F(N) and µ0 ∈/ ext(BF(N)).
Proof. Writeµ0 = 12(µ1+µ2), withµ1, µ2 ∈BF(M). We can find sequences{νni} of finitely supported measures such that µi = limn→∞νni for i = 0,1,2. Let N ={0} ∪(∪i,nsupp{νni}). Note that the canonical inclusion F(N),→ F(M) is an isometry and νni ∈ F(N) for each n, i. Since F(N) is complete, it is a closed subspace of F(M). Thus µ0, µ1, µ2 ∈ F(N) and so µ0 ∈/ext(BF(N)).
Proof of Proposition 5.1. Let mxy be a molecule in M such that [x, y] = {x, y}
and assume that mxy ∈/ ext(BF(M)). By Lemma 5.2, we may assume that M is countable. Write M = {xn : n ≥ 0}. Let {en :n ≥1} be the unit vector basis of `1. It is well known that the map δ(xn) 7→ en for n ≥ 1 defines an isomorphism from F(M) onto `1. Thus {δ(xn) :n ≥1} is a Schauder basis for F(M).
Assume that mxy = 12(µ+ν) for µ, ν ∈BF(M) and write µ=P∞
n=1anδ(xn).
Fix n∈N such that xn∈ {x, y}. Then, there is/ εn >0 such that (1−εn) (d(x, xn) +d(xn, y))≥d(x, y).
Let gn =fxy+εn1{xn}, which is an element of Lip0(M) since M is uniformly discrete. We will show that kgnkL ≤ 1. To this end, take u, v ∈ M, u 6= v.
Since kfxykL ≤ 1, it is clear that |hgn, muvi| ≤ 1 if u, v 6= xn. Thus we may
assume v = xn. Therefore (c) in Lemma 1.1 yields that hfxy, muvi ≤ 1−εn
and so hgn, muvi ≤1. Exchanging the roles of uand v, we get that kgnkL≤1.
Moreover, note that
1 =hgn, mxyi= 1
2(hgn, µi+hgn, νi)≤1
and so hgn, µi = 1. Analogously we show that hfxy, µi = 1. Thus an = h1{xn}, µi = 0. Therefore µ = aδ(x) +bδ(y) for some a, b ∈ R. Finally, let f1(t) := d(t, x)−d(0, x) and f2(t) := d(t, y)−d(0, x). Then kfikL = 1 and hfi, mxyi= 1, so we also have hfi, µi= 1 fori= 1,2. It follows from this that a =−b= d(x,y)1 , that is, µ=mxy. This implies that mxy is an extreme point of
BF(M).
Next we show that preserved extreme points are automatically strongly exposed for uniformly discrete metric spaces. Notice that, contrary to other results in this section, no boundedness assumption is needed.
Proposition 5.3. Let M be a uniformly discrete pointed metric space. Then every preserved extreme point of BF(M) is also a strongly exposed point.
Proof. Let x, y ∈ M such that mxy is a preserved extreme point of BF(M). Assume that mxy is not strongly exposed. By Theorem 4.4 in [12], the pair (x, y) enjoys property (Z). That is, for each n ∈Nwe can find zn ∈M \ {x, y}
such that
d(x, zn) +d(y, zn)≤d(x, y) + 1
nmin{d(x, zn), d(y, zn)}.
Thus,
(1−1/n)(d(x, zn) +d(y, zn))≤d(x, y)
so it follows from condition (ii) in Theorem 2.6 that min{d(x, zn), d(y, zn)} →0.
Since M is uniformly discrete, this means that{zn}is eventually equal to either
x or y, a contradiction.
Aliaga and Guirao proved in [2] that, in the case of compact metric spaces, every molecule which is an extreme point of BF(M) is also a preserved extreme point. However, that result is no longer true for general metric spaces, as the following example shows.
Example 5.4. Consider the sequence inc0given byx1 = 2e1, andxn= e1+(1+
1/n)enforn ≥2, where{en}is the canonical basis. LetM ={0}∪{xn:n∈N}.
This metric space is considered in [2, Example 4.2], where it is proved that the molecule m0x1 is not a preserved extreme point of BF(M). Let us note that this fact also follows easily from Theorem 2.6. Moreover, by Proposition 5.1 we have that m0x1 ∈ext(BF(M)).
On the other hand, if we restrict our attention to uniformly discrete bounded metric spaces satisfying the hypotheses of the duality result, then all the families of distinguished points of BF(M) that we have considered coincide.
Proposition 5.5. Let (M, d) be a uniformly discrete bounded pointed metric space such that F(M) admits a natural predual. Then for µ ∈ BF(M) it is equivalent:
(i) µ∈ext(BF(M)).
(ii) µ∈strexp(BF(M)).
(iii) There are x, y ∈M, x6=y, such that µ=mxy and [x, y] ={x, y}.
Proof. (i) ⇒ (iii) follows from Proposition 4.1. Moreover, (ii)⇒(i) trivially.
Now, assume that µ = mxy with [x, y] = {x, y}. We will show that the pair (x, y) fails property (Z) and thus µ is a strongly exposed point. Assume, by
contradiction, that there is a sequence {zn} inM such that d(x, zn) +d(y, zn)≤d(x, y) + 1
nmin{d(x, zn), d(y, zn)}.
and so
(1−1/n)(d(x, zn) +d(y, zn))≤d(x, y).
The compactness with respect to the w∗-topology ensures the existence of a w∗-cluster pointz of {zn} (M and δ(M) ⊂ F(M) being naturally identified).
Now, by the lower semicontinuity of the distance, we have d(x, z) +d(y, z)≤lim inf
n→∞ (1−1/n)(d(x, zn) +d(y, zn))≤d(x, y).
Therefore, z ∈ [x, y] = {x, y}. Suppose z = x. Denote θ = inf{d(u, v) : u 6=
v}>0. The lower semicontinuity of d yields θ+d(x, y)≤lim inf
n→∞ (1−1/n)(θ+d(y, zn))
≤lim inf
n→∞ (1−1/n)(d(x, zn) +d(y, zn))≤d(x, y),
which is impossible. The case z =y yields a similar contradiction. Thus the
pair (x, y) does not have property (Z).
We now give some examples in which the preduals of F(M) have interesting properties. The first one is a uniformly discrete and bounded metric space M such that F(M) is isometric to a dual Banach space but cannot admit a natural predual. This example comes from [2, Example 4.2] and has already been introduced in Example 5.4.
Example 5.6. Consider the sequence in c0 given by x0 = 0, x1 = 2e1, and xn = e1 + (1 + 1/n)en for n ≥ 2, where {en} is the canonical basis. Let M ={0} ∪ {xn:n∈N}. Then
a) F(M) does not admit any natural predual.
b) The space X ={f ∈Lip0(M) : limf(xn) = f(x1)/2}satisfies X∗ =F(M).
Our Corollary 3.7 guarantees that in order to prove a) it is enough to show that there is no compact topology τ onM such that d isτ-l.s.c. Assume that τ is such a topology. Then the sequence {xn} admits a τ-accumulation point x ∈ M. Since d is τ-l.s.c. we get that x ∈ B(0,1)∩B(x1,1). But this is a contradiction as the latter set is clearly empty.
For the proof of b) we will employ the theorem of Petun¯ın and Pl¯ıˇchko. The space X is clearly a separable closed subspace of F(M)∗. Further, a simple case check shows that for anyx6=y∈M,y 6= 0, the functionf(x) = 0, f(y) =d(x, y) can be extended as an element ofXwithout increasing the Lipschitz norm. Thus since X is clearly a lattice, Proposition 3.4 of [20] shows thatX is separating.
Finally, if f ∈X and
f(xnk)−f(xmk)
d(xnk, xmk) → kfkL
then without loss of generality the sequence {mk} does not tend to infinity.
Passing to a subsequence, we may assume that it is constant, say mk =m for all k ∈N. If {nk} does not tend to infinity, then f(xd(xi)−f(xm)
i,xm) =kfkL for some i6=m. Otherwise, since f ∈X, we have
f(xnk)−f(xm) d(xnk, xm) →
f(x1)
2 −f(m)
d(x1, xm) . So in this case the norm is attained at d(x1
1,xm)(δ(x1)/2−δ(xm))∈BF(M). It follows that every f ∈X attains its norm. Thus by the theorem of Petun¯ın and Pl¯ıˇchko, X∗ =F(M).
Next we show that F(M) can actually have both natural and non-natural preduals.
Example 5.7. Let M = {0} ∪ {1,2,3, . . .} be a graph such that the edges are couples of the form (0, n) with n ≥ 1. Let d be the shortest path distance on M. Then it is obvious and well known that F(M) is isometric to `1. Moreover F(M) admits both natural and non-natural preduals. Indeed, an example of a natural predual is X = {f ∈Lip0(M) : limf(n) =f(1)} (this is immediate using Corollary 3.7). An example of a non-natural predual is Y ={f ∈Lip0(M) : limf(n) =−f(1)}. We leave to the reader the verification of the hypotheses of the theorem of Petunin and Plichko.
Our last example shows that there are uniformly discrete bounded metric spaces such that their free space does not admit any isometric predual at all.
Such observation is relevant to the open problem whether F(M) has (MAP) for every uniformly discrete and bounded metric space M (see also Problem 6.2
